MATH 1330 Section 2.3 Rational Functions and Their Graphs Vertical Asymptote of Rational Functions The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. Locating Vertical Asymptotes and Holes Factor the numerator and denominator. Look at each factor in the denominator. If a factor cancels with a factor in the numerator, then there is a hole where that factor equals zero. If a factor does not cancel, then there is a vertical asymptote where that factor equals zero. Horizontal Asymptote of Rational Functions The line y = b is a horizontal asymptote of the graph of a function f if f(x) approaches b as x increases or decreases without bound. Notes on Horizontal Asymptotes Horizontal asymptotes really have to do with what happens to the y- values as x becomes very large or very small. If the y-values approach a particular number at the far left and far right ends of the graph, then the function has a horizontal asymptote. Note: A rational function may have several vertical asymptotes, but only at most one horizontal asymptote. Also, a graph cannot cross a vertical asymptote, but may cross a horizontal asymptote. Locating Horizontal Asymptotes Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. To find the equation of the slant asymptote, divide the denominator into the numerator. The quotient is the equation of the slant asymptote. As |x| -> , the value of the remainder is approximately 0. The equation of the slant asymptote will always be of the form y = mx + b. Steps to Graphing a Rational Function 1.Factor numerator and denominator. If a factor in the numerator cancels with a factor in the denominator then there is a hole in the graph when that cancelled factor equals zero. 2. Find the x-intercept(s) by setting the numerator equal to zero. 3. Find the y-intercept (if there is one) by substituting x = 0 in the function. 4. Find a horizontal asymptote, if there is one. If not, then see if it has a slant asymptote and find its equation. 5. Find the vertical asymptote(s), if any, by setting the denominator equal to zero. 6. Use the x-intercept(s) and vertical asymptote(s) to divide the x-axis into intervals. Choose a test point in each interval to determine if the function is positive or negative there. This will tell you whether the graph approaches the vertical asymptote in an upward or downward direction. 7. Graph! Except for the breaks at the vertical asymptotes, the graph should be a nice smooth curve with no sharp corners. Note: It is possible for the graph to cross the horizontal asymptote, maybe even more than once. The method used to figure out graph of a function crosses (and where), set y equal to the value of the horizontal asymptote and then solve for x. Example 6: Does the function cross the horizontal asymptote? Popper #2: EMCF 2 a b. c.